The document discusses a consumer, Webster, who prefers bundles of two goods, emphasizing his strong preference for the first good over the second. It examines his demand and preferences under certain constraints, introducing concepts like utility functions and continuity of preferences. The conclusion asserts that there is no continuous utility function that can represent Webster's preference relation.

1. Consider a coestmer, Wetister, who may spend his income on bamultes of two gooks: first good"
(f) and "aecoed good" (s). Lat ( f , s ) denote basdles of f units of the fint good and a asite of the
wornd pood; both goods are avalable is cuitimusus tinerements. So, the mot of all posible bundis
would be X R + 2 (the neh-argative pquadrant of two dinetisional rot epoce). Welster has uunatal
preferenres; you could say he has a onetrack mind. Condider two hemdies in X 1 , x = ( f 1 , 1 )
and y = ( f 2 , 2 ) . If f 1 > f 2 then z > v rexardlese of the equantities s 1 and s 2 be wants as
muxh of the first goed an breab get But if f 1 = f 2 , he peveres whichever of the baballe provides
him with more of the aqood good. If f 1 f 2 and x 1 x 2 . Wetster in iadiffestat hetwera the two
basulless thes appais, he in coseparing two ideutical basilis aay not be as diwious is that the Z m
owe X is alis strongly manatone and stricify coinec. An informal (tat cunplete) proof is fine. Bat
bark to Wetater's strangencae. To iltatrate further his ecossuricity, aupjoie that be alwass faces
the folloeiag tradget eonstraint: p f f + p n whrre ( p f , p 0 ) are the price of the first good and of
the mexod good, respectively, atud v is hin income. Webnter it a prie-tader in the two goods. (b)
Gives his peeferences, tell ne bow much Welweter would consuse of each good given ( p y , p n ,
y ) . The ielation you derive abowe is an expresion for Wehster's (Marehalliad) demast for each
good - we will tackle Marshallian alemand in more detail lates (or, if you ure froctactinating. by
the weird about Webister 's peeference relation. And there is. His Z p oner X satisfies tationalify,
strong monotonicity, abd exy strict convexity ... bat not centennityl The remaining parts of this
question are meant to walk you through the argumends demosatrating that Webeter's prefatsee
relation is not continuoes. To do this, you will show that there in no continoses utility fubation
that can repreaeat Welister'y joeferenor relation. Let z ( f , s ) X be some abbitrary bunile. Let V (
x ) drnote a utility function eonsistent with Welses's preference relation zo over X . For out
purpons, let a ntility function U be considered n + x lim U ( x 41 ) = U ( x ) (c) Let a , b > 0 .
Which in greates, U ( f + a , s ) of U ( f , n + b ) ? Does the aaseer depend on the relative values
of a and 6 ? Consider two sequenere. The first is { x f n = 1 , where x = f . x + n 1 } . The second
is { x 2 } n = 1 where I ~ n = ( f + n 1 , x ) . Note that both sequences converge to the same
bandle as n , nainely As we will want to use "proof by contradtiction," kit ws isitidly assume that
U(') ts a contimous function; so assume lim n n U ( x n ) = U ( x ) and limen U ( x n ) = U ( x )
(d) Explaio who n + lim U ( 2 n ) > U ( x ) for acy positive integed . Hint, Start by arguing why
V ( F + ) > V ( r ) for any two positise integrs i and f (e) Explais why U ( x 2 ) > n lim U ( e n )
for any positive integer j . Hint: Note that the smuenee { U ( x ) ) is s serictly deellinigg in that U
( x 2 ) > U ( x 2 ) for is y . (f) Now use what you have shoiWm so far to angue that m + lim U ( 2
^ n ) > n lim U ( x n ) and, therefoee, the utility function U reperenetaing Webster's preference
relation Z ove X cannot be contimasus. (Proof by Contracliction) (s) Lastly, ooasider some
atbitrary banalle x . Is there any other bundke x X macle that x f foe Wetoter? Relatt your anserer
to this part with that for (b).